Therefore, through the study of the PHE in both honeycomb lattices [1]} and kagome lattices, we find discontinuous jumps in Chern numbers, which manifest themselves as singularities of the first derivative of the phonon Hall conductivity with respect to the magnetic field. Such associated phase transition is connected with the crossing of band 2 and band 3, which corresponds to the touching between a acoustic band and a optical band. However, we can not observe the similar associated phase transition in triangular lattices because of no optical bands, where the Chern number of each band keeps zero while the phonon Hall conductivity is nonzero because of the nonzero Berry curvatures.
